Finite point configurations and projection theorems in vector spaces over finite fields

Abstract Let $G_1, \ldots , G_k$ be finite-dimensional vector spaces over a prime field $\mathbb {F}_p$ . A multilinear variety of codimension at most $d$ is a subset of $G_1 \times \cdots \times G_k$ defined as the zero set of $d$ forms, each of which is multilinear on some subset of the coordinates. A map $\phi$ defined on a multilinear variety $B$ is multilinear if for each coordinate $c$ and all choices of $x_i \in G_i$ , $i\not =c$ , the restriction map $y \mapsto \phi (x_1, \ldots , x_{c-1}, y, x_{c+1}, \ldots , x_k)$ is linear where defined. In this note, we show that a multilinear map defined on a multilinear variety of codimension at most $d$ coincides on a multilinear variety of codimension $O_{k}(d^{O_{k}(1)})$ with a multilinear map defined on the whole of $G_1\times \cdots \times G_k$ . Additionally, in the case of general finite fields, we deduce similar (but slightly weaker) results.

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The Number of Occurrences of a Fixed Spread among n Directions in Vector Spaces over Finite Fields

Graphs and Combinatorics ◽

10.1007/s00373-012-1242-3 ◽

2012 ◽

Vol 29 (6) ◽

pp. 1943-1949 ◽

Cited By ~ 1

Author(s):

Le Anh Vinh

Keyword(s):

Finite Fields ◽

Vector Spaces

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Restriction Operators Acting on Radial Functions on Vector Spaces over Finite Fields

Canadian Mathematical Bulletin ◽

10.4153/cmb-2014-016-2 ◽

2014 ◽

Vol 57 (4) ◽

pp. 834-844

Author(s):

Doowon Koh

Keyword(s):

Finite Field ◽

Finite Fields ◽

Vector Spaces ◽

Dimensional Vector ◽

Algebraic Varieties ◽

Test Functions ◽

Radial Functions ◽

Zero Radius ◽

Intersection Points ◽

Field Setting

AbstractWe study Lp → Lr restriction estimates for algebraic varieties V in the case when restriction operators act on radial functions in the finite field setting. We show that if the varieties V lie in odd dimensional vector spaces over finite fields, then the conjectured restriction estimates are possible for all radial test functions. In addition, assuming that the varieties V are defined in even dimensional spaces and have few intersection points with the sphere of zero radius, we also obtain the conjectured exponents for all radial test functions.

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